Andreev et al.~\cite{ABCR97} gave constructions of Booleanfunctions (computable by polynomial-size circuits) with large lower boundsfor read-once branching program (1-b.p.'s): a function in P with the lowerbound 2^{n-\polylog(n)}, a function in quasipolynomial time with the lowerbound 2^{n-O(\log n)}, and a function in LINSPACE with the lower bound2^{n-\log n -O(1)}. We point out alternative, much simpler constructionsof such Boolean functions by applying the idea of almost k-wiseindependence more directly, without the use of discrepancy set generatorsfor large affine subspaces; our constructions are obtained byderandomizing the probabilistic proofs of existence of the correspondingcombinatorial objects. The simplicity of our new constructions alsoallows us to observe that there exists a Boolean function in AC^0[2](computable by a depth 3, polynomial-size circuit over the basis{\wedge,\oplus,1}) with the optimal lower bound 2^{n-\log n -O(1)} for1-b.p.'s.

Andreev et al.~\cite{ABCR97} give constructions of Booleanfunctions (computable by polynomial-size circuits) that require largeread-once branching program (1-b.p.'s): a function in P that requires1-b.p. of size at least $2^{n-\polylog(n)}$, a function in quasipolynomialtime that requires 1-b.p. of size at least $2^{n-O(\log n)}$, and afunction in LINSPACE that requires 1-b.p. of size $2^{n-\log n -O(1)}$. Wepoint out alternative, much simpler constructions of such Booleanfunctions by applying the idea of almost $k$-wise independence moredirectly, without the use of discrepancy set generators for large affinesubspaces. Our constructions are obtained by derandomizing theprobabilistic proofs of existence of the corresponding combinatorialobjects.